# Physics in Linear Algebra

Talking about physical phenomena related to a particular field of mathematics can be interesting for students and might further motivate their study of the subject. For instance, there are interactions between physics and linear algebra, and discussion of these interactions could simultaneously motivate physics students to study the underlying mathematics and inspire mathematics students to appreciate the physical applications.

Question: What are good examples of physical phenomena related to linear algebra that are comprehensible by undergraduate students? Could these be feasibly incorporated into a linear algebra course?

• I think the existence of principlal axes for the moment of inertia is interesting: en.wikipedia.org/wiki/Principal_axis_(mechanics)#Principal_axes – Brian Rushton Apr 2 '14 at 1:12
• @BrianRushton Thank you very much. Would you please add it as an answer with more explanation? – user230 Apr 2 '14 at 1:14
• I added to the motivation of the question, edited some grammar, and added the tag "physical-sciences". There's currently a meta thread about this tag, as well: meta.matheducators.stackexchange.com/questions/249/… – Brendan W. Sullivan Apr 2 '14 at 3:25
• What do you mean by "no presumed advanced background in physics?" Does advanced mean more than a freshman survey? Don't a lot of people take linear algebra without having had any exposure to physics whatsoever? – Ben Crowell Apr 2 '14 at 4:55
• @brendansullivan07 Thanks for your edit. – user230 Apr 2 '14 at 9:48

Here is a list of how I actually used linear algebra as a physics major. The general techniques are

• Change of basis (including calculating reciprocal vectors, which in turn require calculating determinants)
• Projections onto subspaces
• Finding the eigenvalues and eigenvectors (including calculating determinants)

Some examples

1. calculating the intersections of lines and planes
2. Finding the axis of rotation (eigenvector)
3. Calculating the rotation matrix (Euler angles)
4. calculating the volume of a crystal cell (determinant)
5. calculating a cross product as a determinant in R3 (this is a neat trick as the cross product is not always used to find the normal to a plane, but if you present it as the area of an oriented 2-blade then it generalizes Rn)
6. calculating the coefficients of the metric tensor in a different coordinate system. After calculating the partial derivatives, the metric tensor can be transformed by a similarity transformation.
7. calculating the reciprocal basis vectors. Students may be familiar with orthonormal systems, but lots of questions arise when dealing with curvilinear systems. Writing the gradient operator in terms of the reciprocal basis vectors clears up everything. Reciprocal vectors are also used in solid state (crystals have natural axes which may not be orthogonal and natural length scales which may be different along each axis).
8. Projecting vectors onto orthonormal bases (easy with geometric vectors, but not recognized when generalized to functions)
9. Projecting vectors onto orthogonal bases. Students are familiar with spherical/cylindrical coordinates using orthonormal basis vectors. But calculating gradient, divergence, curl, and laplacian, are all magic. It is a good way to introduce reciprocal vectors and less familiar orthogonal coordinate systems. Make sure to generalizing from geometric vectors to non-normalized eigenfunctions.
10. Projecting vectors onto non-orthogonal bases. This requires calculating reciprocal vectors. This is useful in solid-state, as well as special and general relativity. It also encourages students to make up coordinate systems, like ellipsoidal coordinates z=a r cos q, x=b r sin q cos f, y=c r sin q sin f.
11. calculating the principal moments and principal axes of the moment of inertia.
12. calculating the (eigen)modes and modal frequencies of vibrating systems. This deals with Fourier series and orthogonal functions.
13. expanding a wavefunction in terms of stationary states (eigenfunctions). Calculating the probability of a measuring a particular eigenvalue. Various problems can help make the connection between Fourier series, legendre polynomials and spherical harmonics as just various orthogonal bases.
14. Multipole expansions in various fields (acoustics, e&m). Yet another example of projecting onto bases.
• (+1) Thanks for your useful answer. Welcome to MESE, Timothy! – user230 Apr 3 '14 at 22:04
• Something I realized later are the matrix representations of Lie groups... In particular the galilean group and the special euclidean group. We know that rotations are represented by matrices, but I was told translations don't work like that. However, adding an extra dimension or two to your matrices DOES allow translations and galilean boosts. – Timothy Wofford Apr 4 '14 at 7:15
• Also numerical computations (finite elements, finite difference, etc.) all involve working with matrices. – Timothy Wofford Apr 4 '14 at 7:16
• Change of basis is a big one. I like to play with orbits. Things are a whole lot simpler if I set orbital plane as the xy plane, that way it's no longer necessary to mess with (x,y,z) co-ordinates, (x,y) coordinates do just fine. And this can be accomplished using a rotation matrix. – HopDavid Apr 4 '14 at 23:19
• What the heck “reciprocal vectors” are? BTW, won’t check maths in the posting because of virtually non-existent typesetting. – Incnis Mrsi Aug 24 '15 at 19:01

I have two good examples, they are somewhat similar, but one is solved more analytically, and the other more computationally:

# Masses on a spring:

From classical mechanics: a mass on a spring with one end fixed can be described by the classical equation:

$\mathbf{F} = m\ddot{\mathbf{x}} = -k\mathbf{x}$

If we have two masses on a spring, (in 1d)

$m_1 \ddot{x}_1 = -k(|x_1 -x_2| - \ell_{\text{eq}})$

$m_2 \ddot{x}_2 = -k(|x_1 -x_2| - \ell_{\text{eq}})$

When you get to more springs, this is more easily posed as a linear algebra problem:

$\mathbf{M} \ddot{\mathbf{x}} = \mathbf{K}\mathbf{x}$

Where now, $\mathbf{M}$, and $\mathbf{K}$ are matrices describing the masses of the different objects and the spring constants between them respectively, and $\mathbf{x}$ describes the positions of the various objects (with the equilibrium distance $\ell$ subsumed into it).

If you look at this as a simple eigenvalue equation:

$\ddot{\mathbf{x}} = \mathbf{M}^{-1}\mathbf{K} \mathbf{x}$

The eigenvectors (remember that $x$ is really time dependent), describe the normal modes of the system, and the eigenvalues are the frequencies of these modes. Demonstrating this with a system of 3 or so masses isn't THAT hard, and is very cool.

This equation is so fundamental to physics that it can really drive home how ridiculously important linear algebra is to Physicists. Unfortunately, describing what the wave function represents can be difficult. The best way is to describe the square of the wave function as a probability distribution. Thankfully as well, the steady state problem is real, and you don't have to deal with complex numbers. We can write down the Schrödinger equation as follows:

${\cal H}\Psi(t) = \left(-\frac{\hbar^2}{2m} \boldsymbol{\nabla} + V(\mathbf{x}) \right) \Psi(t) = -i \hbar \partial_t \Psi(t)$

From here, we can choose a basis by separating out the time dependence:

$\left(-\frac{\hbar^2}{2m} \boldsymbol{\nabla} + V(\mathbf{x}) \right) \Psi_e = e \Psi_e$

And then we can write:

${\cal H}\Psi_e = e \Psi_e$

Where ${\cal H}$ is the finite difference expression of $\Psi$ on a grid. For example, for the 1D problem with a simple harmonic oscillator:

${\cal H}\Psi = -\frac{\hbar^2}{2m h^2} \left(\Psi(x_{i-1}) - 2\Psi(x_{i} + \Psi(x_{i+1}) \right) + \frac{k}{2} x_i^2 \Psi(x_{i})$

Where $i$ is an index on the evenly spaced discretized grid.

Then you can find the eigenvalues and eigenvectors (unfortunately, for anything on a reasonable grid, this requires too many points to do by hand). These eigenvalues describe the steady state energies of the system! And the eigenvectors describe the steady state wave functions! Interestingly enough, this particular problem describes vibrational levels in molecules remarkably well (especially for the lowest energy states).

• (+1) You mentioned very nice points. Welcome to MESE, Andrew! – user230 Apr 3 '14 at 22:06

This should rivet students' attention!

See the MathOverflow question, "The “Dzhanibekov effect”--an exercise in mechanics or fiction? Explain mathematically a video from a space station," and especially the beautifully informative response by Terry Tao here.

It is also known as the Tennis racket theorem (Wikipedia). (The remarkable) video URL here: • (+1) It is really interesting. Thank you very much. – user230 Apr 3 '14 at 23:28

In classical mechanics, the motion of a rigid body can be described completely by the motion of the center of mass and by the rotations around the center of mass.

As opposed to linear velocity and momentum, the angular velocity does not always point in the same direction as the angular momentum. As mentioned on this site, this can lead to problems as it creates wobbling in rotating machinery.

The principal axes theorem says that each rigid object has three perpendicular axes about which you can rotate the object and have the angular momentum match up with the angular velocity. These are the best axes to rotate an object by.

The proof is nothing but the fact that a symmetric matrix can be diagonalized by an orthogonal change of basis.